Accurate and efficient algorithms for boundary element methods in electromagnetic scattering: a tribute to the work of F. Olyslager

Boundary element methods (BEMs) are an increasingly popular approach to model electromagnetic scattering both by perfect conductors and dielectric objects. Several mathematical, numerical, and computational techniques pullulated from the research into BEMs, enhancing its efficiency and applicability. In designing a viable implementation of the BEM, both theoretical and practical aspects need to be taken into account. Theoretical aspects include the choice of an integral equation for the sought after current densities on the geometry's boundaries and the choice of a discretization strategy (i.e. a finite element space) for this equation. Practical aspects include efficient algorithms to execute the multiplication of the system matrix by a test vector (such as a fast multipole method) and the parallelization of this multiplication algorithm that allows the distribution of the computation and communication requirements between multiple computational nodes. In honor of our former colleague and mentor, F. Olyslager, an overview of the BEMs for large and complex EM problems developed within the Electromagnetics Group at Ghent University is presented. Recent results that ramified from F. Olyslager's scientific endeavors are included in the survey

A Study of Speed of the Boundary Element Method as applied to the Realtime Computational Simulation of Biological Organs

In this work, possibility of simulating biological organs in realtime using the Boundary Element Method (BEM) is investigated. Biological organs are assumed to follow linear elastostatic material behavior, and constant boundary element is the element type used. First, a Graphics Processing Unit (GPU) is used to speed up the BEM computations to achieve the realtime performance. Next, instead of the GPU, a computer cluster is used. Results indicate that BEM is fast enough to provide for realtime graphics if biological organs are assumed to follow linear elastostatic material behavior. Although the present work does not conduct any simulation using nonlinear material models, results from using the linear elastostatic material model imply that it would be difficult to obtain realtime performance if highly nonlinear material models that properly characterize biological organs are used. Although the use of BEM for the simulation of biological organs is not new, the results presented in the present study are not found elsewhere in the literature.Comment: preprint, draft, 2 tables, 47 references, 7 files, Codes that can solve three dimensional linear elastostatic problems using constant boundary elements (of triangular shape) while ignoring body forces are provided as supplementary files; codes are distributed under the MIT License in three versions: i) MATLAB version ii) Fortran 90 version (sequential code) iii) Fortran 90 version (parallel code

Efficient sum-of-exponentials approximations for the heat kernel and their applications

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order $O(\log(\frac{T}{\delta}) (\log(\frac{1}{\epsilon})+\log\log(\frac{T}{\delta})))$ for any x\in\bbR and $\delta \leq t \leq T$, where $\epsilon$ is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only $O(\log^2(\frac{T}{\delta}))$ terms for fixed accuracy $\epsilon$. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For $N_S$ points in the spatial discretization and $N_T$ time steps, the cost is $O(N_S N_T \log^2 \frac{T}{\delta})$ in terms of both memory and CPU time for fixed accuracy $\epsilon$. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.Comment: 23 pages, 5 figures, 3 table